Find the trigonometric integral. ∫tan2xsec3xdx

Question

Find the trigonometric integral.  ∫tan2xsec3xdx

movingsupplyw1 · Accepted Answer

Step 1
Integral of a function is the anti derivative of the function, meaning area of the graph of the function , under the given interval. We have standard integral forms and formulas for finding the same.
We can break the integral into sum of integrals and make proper substitution to find the integrals.
Step 2
We can simplify the above integral as,

\int \frac{\tan^{2} x}{\sec^{3} x} d x = \int \frac{\sin^{2} x}{\cos^{2} x \cdot \frac{1}{\cos^{3} x}} d x

= \int \sin^{2} x \cos x d x

Taking

\sin x

as t, we get

\cos x d x = d t

, making this replacement, new integral becomes,

= \int t^{2} d t

= \frac{t^{3}}{3} + c

, replacing t with original value,

= \frac{\sin^{3} x}{3} + c

Terry Ray · Accepted Answer

∫tan2(x)sec3(x)dx  Rewrite / simplify using the definition of trigonometric / hyperbolic functions:  =∫cos⁡(x)sin2(x)dx  Substitution u=sin⁡(x)⇒dudx=cos⁡(x)⇒dx=1cos⁡(x)du:  =∫u2du  Integral of a power function:  ∫undu=un+1n+1 at n=2:  =u33  Reverse replacement u=sin⁡(x):  =sin3(x)3  ∫tan2(x)sec3(x)dx  =sin3(x)3+C

nick1337 · Accepted Answer

∫tan⁡(x)2sec⁡(x)3dx Transform the expressions ∫(sin⁡(x)cos⁡(x))2(1cos⁡(x))3dx Use the properties of exponents ∫sin⁡(x)2cos⁡(x)21cos⁡(x)3dx Simplify the expression ∫sin⁡(x)2cos⁡(x)dx Transform the expression ∫t2dt Evaluate the integral t33 Substitute back sin⁡(x)33 Add C Solution sin⁡(x)33+C

Find the trigonometric integral. \int \frac{\tan^{2}x}{\sec^{3}x}dx

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